Assignment of Series 2 of Year 21

You travel in fast train and are looking outside of the open window. Three windows in front of you someone spits a chewing-gum. How long do you have to get back to coupe to avoid contact with chewing-gum? The chewing-gum is spherical shape and was not thrown out, but just laid in air flow.

Calculate a slope of front glass of a car, so that water drops at speed 80 km ⁄ h do not run off, but to the sides. Verify that you results is compatible with reality. What else influence the slope of front window?

Wine makers and truck drivers know how to move liquid from heavy containers. Winemaker Ignac wants to move wine from one container to another. First he puts empty container on the ground and full to the level $Δ$. Then both containers are connected by a tube and with help of little suction wine starts to flow to bottom container. How long it will take? Assume, that both containers are same cylinders of radius $R$ and height $H$.

Two identical charges are at the end of stiff non-conductive rod. What power will be needed to rotate rod at constant angular speed with axis going through the middle of the rod? The friction is negligible

A submarine has dived into deep ocean in Marian ditch and released a bubble of air. It started to go to surface. However, when you calculate, using gas equation, density of the air, you will see that it is bigger than density of water. Is it possible?

If you agree, explain your answer. If you do not agree, calculate what will be parameters of the bubble (mainly density).

Verify experimentally following hypothesis: the rotation of Earth causes water on north hemisphere to swirl to right, on south hemisphere to left. For your conclusion to have relevance, enough number of measurements must be done at different conditions.

Uranium storage

Very important question is storing of radioactive waste. Usually it is stored in cylindrical containers immersed in water, which keeps the surface at constant temperature 20 °C. Your task is to find the temperature distribution inside containers of square base of edge length 20 cm. Container is relatively long, therefore just temperature distribution in horizontal cross section is of interest. Uranium will be in block of square base of edge 5 cm. From the experience with cylindrical capsules we know, that it will have constant temperature of about 200 °C.

Heating wire

Lets have a long wire of circular cross section and radius $r$ from a material of heat conductivity $λ$ and specific conductivity $σ$. Then a electric field is applied. Lets the electric field inside the wire is constant and parallel with the axis of the wire and the strength is $E$. Then the current through wire will be $j=σE$ and will create Joule's heat with volume wattage $p=σE$.

Because the material of the wire has non-zero temperature conductivity, some equilibrium gradient of temperature will form. The gradient fulfills Poisson's equation $λΔT=−p$. Assume, that the end of wire is kept at temperature $T_{0}$. This gives a border condition needed to solve the equation. Due to symmetry we can take into account only two dimensions: on cross section of wire (temperature will be independent of shift along the axis of symmetry). Now it is easy to solve the problem with methods described in text.

However, we will make our situation little bit more complex and will assume, that specific electrical conductivity $σ$ is function of temperature. So we will have a equation of type Δ$T=f(T)$.

Try to solve this equation numerically and solve it for some dependency of conductivity on temperature (find it on internet, in literature of just pick some nice function) and found temperature profile in wire profile. Try to change intensity of electric field $E$ and plot volt-amper characteristics, you can try more than one temperature dependency. $σ(T)$ (e.g. semiconductor which conductivity increase with temperature, or metal, where conductivity is decreasing) etc.

Do not limit your borders, we would be glad for any good idea.

Capacity of a cube

Calculate capacity of ideally conductive cube of edge length 2$a$ (2Ax2Ax2A). If you think, it is simple, try to calculate for cuboid (AxBxC) or other geometrical shapes.

Hint: Capacity is a ration of the charge on the cube to the potential on the surface of cube (assuming that the potential in infinity is zero). Problem can be solved by selecting arbitrary potential of cube and solving Laplace equation Δ$φ=0$ outside of the cube and calculating total charge in cube using Gauss law. E.g. calculating intensity of electrical field and derivating potential and calculation of flow through nicely selected surface around the cube.

Final solution is finding a physical model, its numerical solution and realization on computer. More points you will get for deeper physical analysis and detailed commentar. For algorithm you can also get extra points.